Elasticity of two-dimensional filaments with constant spontaneous curvature

We study the mechanical property of a two-dimensional filament with constant spontaneous curvature and under uniaxial applied force. We derive the equation that governs the stable shape of the filament and obtain analytical solutions for the equation. We find that for a long filament with positive initial azimuth angle (the azimuth angle is the angle between x axis and the tangent of the filament) and under large stretching force, the azimuth angle is a two-valued function of the arclength, decreases first, and then increases with increasing arclength. Otherwise, the azimuth angle is a monotonic function of arclength. At finite temperature, we derive the differential equation that governs the partition function and find exact solution of the partition function for a filament free of force. We obtain closed-form expressions on the force-extension relation for a filament under low force and for a long filament under strong stretching force. Our results show that for a biopolymer with moderate length and not too small spontaneous curvature, the effect of the spontaneous curvature cannot be replaced by a simple renormalization of the persistence length in the wormlike chain model.